Discrete Mathematics with Graph Theory (3rd Edition) 373

Discrete Mathematics with Graph Theory (3rd Edition) 373 -...

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Section 13.3 371 21. [BB] This means we can delete the edge joining "Econ" to "Stat," so "Stat" can be colored 3, X(Q) = 3 and three time periods now suffice. 22. Put two varieties into different boxes if one attacks the other. Make a graph whose vertices are varieties and where an edge indicates that the variety represented by one end vertex attacks the other. Since X(Q) = 4, as shown, four is the minimum number of boxes required. 23. Two children must be put in different rooms if one fights with the other. We make a graph 9 whose vertices represent the 15 children. Two vertices are joined by an edge if the corresponding children fight. The graph contains K4 (consider vertices 3, 4, 5 and 8) so at least four colors are required. We show a 4-coloring, so we conclude that X(Q) = 4. The center can get away with four rooms.
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Unformatted text preview: 24. Make a graph where vertices correspond to cities and an edge signifies that the corresponding cities are within 150 km and, therefore, must be assigned different channels. The number of channels required is the chromatic number of the graph which, in this problem, is five, as shown. (Note that the graph contains K 5 ; vertices A, B, D, F and H.) 25. No new channel is required since the chromatic number of the new graph is still five. City J can be assigned the same channel as E and F. 4 F B 1 26. (a) [BB] The dual of the cube is the octahedron whose graph appears in Fig. 13.5. (b) The dual of the dodecahedron is the icosahedron whose graph appears in Fig. 10.19. 27. [BB] Yes, it is. Both the tetrahedron and its dual are K4. Exercises 13.3 1. (a) [BB] 36 nodes, 14 grid segments, 6 nets. (c) [BB]...
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